I'm not really a pro at puppet building, I'm much better at performing.
Personally, I use hot glue as much as I can (most recommend contact cement, but hot glue is easier to come by) when it comes to overall construction, leaving sewing to the fabrication; when it comes to the fabric for the edges of the mouth, I usually trim it, then roll it up slightly to give it that rounded look, then glue it to the mouth - a little trick I picked up from a Frenchman who shared how he built replicas of the Avenue Q puppets on YouTube.
That's what I do, but again, I'm not quite that good at the building aspect of puppets myself... let's see what others, like Gordon, have to say about it...